Examining the spatial distribution pattern and optimum sample size for monitoring of the white mango scale insect, Aulacaspis tubercularis (Newstead) (Hemiptera: Diaspididae) on certain mango cultivars

International journal of Horticulture, Agriculture and Food science(IJHAF)
Vol-4, Issue-3, May-Jun, 2020
https://dx.doi.org/10.22161/ijhaf.4.3.4
ISSN: 2456-8635
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Open Access
Examining the spatial distribution pattern and
optimum sample size for monitoring of the white
mango scale insect, Aulacaspis tubercularis
(Newstead) (Hemiptera: Diaspididae) on certain
mango cultivars
Moustafa M S Bakry1
and Nagdy F Abdel-Baky2*
1
Scale Insects and Mealybugs Research Dept., Plant Protection Research Institute, A.R.C, Dokii, Giza, Egypt.
1
md.md_sabry@yahoo.com
2
Economic Entomology Dept., Fac. Agric., Mansoura University, Mansoura-35516 Egypt. *Department of Plant Production and Protection,
College of Agriculture and Veterinary Medicine, Qassim University, P.O. Box 6622, Buraidah 51452, Qassim, Saudi Arabia.
2
nafabdel2005@yahoo.com
Abstract— The white mango scale insect, Aulacaspis tubercularis (Newstead) (Hemiptera: Diaspididae) is
one of the most destructive pests of mango trees in Egypt. The main objective of the present work is to
estimate the spatial distribution pattern and minimum sample size for monitoring populations of A.
tubercularis on six different cultivars of mango through the two successive years of 2017/2018 and
2018/2019 at Esna district, Luxor Governorate, Egypt. Data on the indices of distribution and Taylor’s and
Iwao’s regression analyses indicate significant aggregation behaviour during each year in all the tested
cultivars of mango trees, that may be caused by environmental heterogeneity. The regression models of
Taylor’s power law (b) and Iwao’s patchiness (β) were both significantly >1, indicating that A.
tubercularis had an aggregation distribution with a negative binomial distribution during each year in all
the tested mango cultivars. The Iwao regression coefficients were used to determine the optimum sample
size required to estimate populations at three fixed precision levels. The optimum size decreased with
increased density in all levels of precision (5, 10 and 15%) in all tested mango cultivars. These can be
deployed to develop a sampling plan to estimate the population density accurately. Results suggesting that
the optimum sample size was flexible and the precision levels of 5 and 10% were suitable for ecological or
insect behavioral studies of A. tubercularis where a higher level of precision is required, whereas, for pest
management programs, a 15% level would be acceptable. Furthermore, the distribution, different mango
cultivars, and sampling protocol presented here could be used as a tool for future research on pest
management methods for this pest.
Keywords—Aulacaspis tubercularis, population density, spatial distribution, sample size, mango
cultivars.
I. INTRODUCTION
Mango trees (Mangifera indica L.) are subjected to
infestation by different pests. Among several pests, infesting
mango trees (Mangifera indica L.), the white mango scale
insect, Aulacaspis tubercularis (Newstead) (Hemiptera:
Diaspididae) is one of the most destructive pests of mango
trees in Egypt. This pest injures the shoots, twigs, leaves,

ISSN: 2456-8635
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branches, and fruits by sucking the plant sap with the
mouthparts, causing thereafter deformations, defoliation,
drying up of young twigs, dieback, poor blossoming, death of
twig by the action of the toxic saliva and so affecting the
commercial value of fruits and their export potential especially
to late cultivars where it causes conspicuous pink blemishes
around the feeding sites (Nabil et al., 2012 and Sayed, 2012).
In nurseries, severe early infestation retards growth. Young
trees are particularly vulnerable to excessive leaf loss and
death of twigs, during hot dry weather (El-Metwally et al.,
2011). The heavily infested premature fruits dropping and the
mature fruits became small in size with lacking juice and were
rotted and unfit for commercial use, in addition to reductions
in quality and quantity of produced fruits (Bakr et al., 2009
and Abo-Shanab, 2012).
Spatial distribution is one of the most characteristic
properties of insect populations; in most cases, it allows us to
define them and is an important characteristic of ecological
communities (Debouzie and Thioulouse 1986). No field
sampling can be efficient without understanding the
underlying spatial distribution of the population (Taylor
1984). An understanding of the spatial distribution (i.e.
regular, random, or aggregated) of populations provides useful
information, not only for theoretical population biology but
also for field monitoring programs, especially sequential
sampling (Feng et al., 1993 and Binns et al., 2000). A
reliable sampling program for estimating the population
density should include a proper sampling time (date of
sampling), sampling unit, and the number of samplings in
which the determination of spatial distribution is crucial
(Pedigo, 1994; Southwood and Henderson, 2000).
The regression models of Taylor (1961) and Iwao (Iwao
and Kuno, 1968) depend on the relationship between the
sample mean and variance of insect numbers per sampling unit
through time, and can provide a stable relationship from one
year to another, based only on the observed sampling mean
(Bisseleua et al., 2011).
Knowledge of the spatial distribution of an insect is
central to the design of any management program, important
for understanding the bioecology of species, and forms the basis
for developing a sampling protocol (Wearing, 1988; Binns et
al., 2000 and Cho et al., 2001).
No information is available in the literature regarding the
spatial distribution of A. tubercularis. Therefore, the present study
was undertaken to estimate the optimum sample size for this pest
on mango trees. The results of this research can be used to draft
monitoring methods for this pest and ultimately to establish pest
management program strategies for A. tubercularis.
II. MATERIALS AND METHODS
Study area
This study was conducted in a private mango orchard,
Mangifera indica L., located in Esna district, Luxor
Governorate, Egypt. The mango cultivars were Goleck, Balady,
Ewaise, Zebda, Sediek and Hindi Bisinnara.The orchard was at
an altitude of 99 m a.s.l., a latitude and longitude of 25.67° N
and 32.71° E, respectively, and was sampled twice monthly
from the beginning of March 2017 until mid-February 2019.
Four mango trees from each cultivar, almost uniform and
similar in age (10 years-old), size, height, vegetative growth,
received the same horticultural practices (i.e. irrigation,
fertilisation, and pruning), were selected and labeled. These
randomly chosen mango trees did not receive any pesticidal
control before and during the period of the study. Regular
bimonthly samples of 20 leaves per tree were randomly picked
from the terminal shoots of tree. Every sample was placed in a
polyethylene bag, all samples were transferred to the
laboratory for inspection using a stereo-microscope.
The total numbers of live insects (nymphs and adult
females) on the upper and lower surfaces of the mango tree
leaves were counted and recorded, linked to the inspection
date, and presented as mean number of individuals per leaf ±
standard error (SE), to express the population size of pest.
General sampling method:
All sampling was conducted from 23040 leaves on 48
dates over 2-year period, i.e. 4 trees × 20 leaves × 6 cultivars ×
48 dates. Samples were frozen to preserve them for
subsequent processing.
The data obtained were statistically analyzed using the
analysis of variance. The means were compared according to
Duncan’s Multiple Range Test (Duncan, 1955) and Least
Significant Difference test (LSD) at the 5% level was used to
determine the significance among means of cultivars, was
carried out by computer (MSTATC Program software,
1980).
Spatial distribution:
The spatial distribution among the sample units was
determined using twenty two distribution indices and two
regression methodologies, namely Taylor’s(1961) and Iwao’s
(Iwao and Kuno, 1968).

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  2
. . ( / ) 100
( / ) 1D M
RV SE X
I S X


 
 
Distribution indices:
Several estimates are based on sample means and
variances, such as index of dispersion, clumping, crowding
and Green’s index (Green, 1966).
- Mean ( X ): the mean number of individuals as a general
average per leaf during the whole year.
- Range of means of a population: The difference between the
maximum mean number of a population and the minimum for
the whole year was calculated by applying the following
equation:
- Range of Density (R) = Population density maximum –
Population density minimum during the entire year.
- Variance (S2
), standard deviation (S), standard error (SE) and
median (Me) for samples were determined.
- Coefficient of variance (C.V.): To assess the fidelity of
sampling, the coefficient of variation values for the studied
years were compared.
Where, S is the standard deviation of the mean and X is
the mean of population.
- Relative Variation (R.V.) is employed to compare the
efficiency of various sampling methods (Hillhouse and Pitre,
1974). The relative variation for the studied years was
calculated as follows:
Where, SE is the standard error of the mean and X is the
mean of population.
- Variance to mean ratio (S2
/ X ):
The simplest approach used for determining the insect
distribution was variance to mean ratio suggested by Patil and
Stiteler (1974). The value of variance-to-mean is one for
‘Poisson’ distribution, less than one for positive binomial and
more than one for negative binomial distribution. Dispersion
of a population can be classified through a calculation of the
variance-to-mean ratio; namely: S2
/X = 1 random distribution,
< 1 regular distribution, and > 1 aggregated distribution
(where, S2
= sample variance; X = mean of population).
- Index of Lewis (IL):
Lewis index was also calculated as per the formula given
hereunder to determine the dispersion of P. oleae
The value of this index revealed >1 contagious; <1:
regular and =1 random distribution.
- Cassie index (Ca):
The spatial distribution pattern is aggregative, random and
uniform when Ca>0, Ca = 0 and Ca<0, respectively (Cassie,
1962).
- The K value of negative binomial distribution:
The parameter K of the negative binomial distribution is
one measure of aggregation that can be used for insect species
having clumped or aggregated spatial pattern. When K values
are low and positive (K< 2), they indicate a highly aggregated
population; K values ranging from 2 to 8 indicate moderate
aggregation; and values higher than 8 (K> 8) indicate a
random population (Southwood, 1995). The K values were
calculated by the moment's method (Costa et al., 2010), and
given by:
Departure from a random distribution can be tested by
calculating the index of dispersion (ID), where, n: denotes the
number of samples:
ID is approximately distributed as x2
with n-1 degrees of
freedom. Values of ID which fall outside a confidence interval
bounded with n-1 degrees of freedom and selected probability
levels of 0.95 and 0.05, for instance, would indicate a
significant departure from a random distribution.
This index can be tested by Z value as follows:
v = n - 1
If 1.96 ≥ Z ≥ -1.96, the spatial distribution would be
random, but if Z < -1.96 or Z > 1.96, it would be uniform and
aggregated, respectively (Patil and Stiteler, 1974).
- Index of mean clumping (IDM) (David and Moore, 1954):
The David and Moore index of clumping values increase
with increasing aggregation. If the index value = 0, the
distribution is random, positive value for negative binomial
100.. 
X
S
VC

 XSIL
2
22
/)( XXCa S 
)12(2  DIZ
XSnID
/)1( 2

  2
. . ( / ) 100
( / ) 1D M
RV SE X
I S X


 
 

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(aggregated) and negative value for positive binomial
(regular).
- Lloyd’s mean crowding (
*
X ):
Mean crowding (
*
X ) was proposed by Lloyd to indicate
the possible effect of mutual interference or competition
among individuals. Theoretically, mean crowding is the mean
number of other individuals per individual in the same
quadrate:
As an index, mean crowding is highly dependent upon
both the degree of clumping and population density. To
remove the effect of changes in density, Lloyd introduced the
index of patchiness, expressed as the ratio of mean crowding
to the mean. As with the variance-to-mean ratio, the index of
patchiness is dependent upon quadrate size (Lloyd, 1967).
- Index of patchiness (IP): is dependent upon quadrate size.
If IP = 1 random, < 1 regular and > 1 aggregated
- Green’s index (GI):
This index is a modification of the index of cluster size
that is independent of n (Green, 1966).
If GI > 0 or positive values are indicative of aggregation
dispersion, GI < 0 or negative values indicative of uniformity
or regular dispersion, and GI = 0 or negative values closer to 0
indicate randomness.
- To evaluate temporal changes in spatial pattern of P. oleae
population during the studied years, an aggregation index (1/k)
(Southwood and Henderson, 2000) was used.
It was calculated using the following formula:
where: 1/k is aggregation index or Cassie’s index C and ( )
is Lloyd’s patchiness index. The values of 1/k< 0, = 0, and > 0
represent regularity, randomness, and aggregation of the
population in spatial pattern, respectively (Feng and
Nowierski, 1992).
- Analysis of causes of aggregation:
The population aggregations mean (λ) (Blackith, 1961)
was used to analysis the causes for the insect population being
in an aggregated state, and was calculated as follows:
Where, у equals to X2
0.5 when the value of the degree of
freedom is 2K. The aggregation of insect individuals is caused
by environmental factors when λ < 2; on the other hand, if λ
>2, the phenomenon is caused by aggregation behavior or the
aggregation behavior works in combination with the
environment.
Regression methods:
Taylor’s power law
The sampling distributions for the total live stages of A.
tubercularis were modeled using Taylor’s power law (TPL)
(1961). A power law function can be used to model the
relationship between mean and variance as:
2
S a
b
X
where, S2
is the variance, X is the sample mean, and a is the
scaling factor related to sample size. The b measures the
species aggregation, and when b = 1, b<1, and b > 1, the
distribution is random, regular, and aggregated, respectively.
Using a log transformation, one can estimate the
coefficients with linear regression as:
2
log( ) log(a) blog( )s X 
where, a and b are the parameters of the model, estimated by
linearising the equation after a log–log transformation
(Taylor, 1984).
Iwao’s patchiness regression (IPR):
Iwao’s regression method was used to quantify the
relationship between the mean crowding index (
*
X ) and mean
population density (X ) Lloyd’s (1967) and using the
following equation:
*
a XX  
where, α is the index of basic contagion and indicates the
tendency of crowding (positive) or repulsion (negative), ifa =
0 indicates single individuals, a colony (a > 0) or a negative
association of individuals and (0 > a > -1)is the basic
component of the distribution. The slope (β) reflects the
distribution of the population in space and is interpreted in the
same manner as b of the TPL (Iwao and Kuno, 1968).
The goodness of fit for both models was evaluated using
the coefficients of determination (R2
) and multiple
correlations.
A Student’s t-test can be used to determine whether the
colony is composed of single individuals and whether the
]1)/[( 2


XSXX
)/( XXI P


)1/(]1)/[( 2
 nXSGI
1)/(/1 

XXk
XX/

  km 2/

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1 2
2 2
1 2( )
slope
b b
t
SEb SEb

 

1 2
2 2
1 2( )
slopet
SE SE
 
 

 

colonies are dispersed randomly (Sedaratian et al., 2010 and
Moradi-Vajargah et al., 2011).
Test b = 1, ( 1) / bt b SE  and Test β = 1,
( 1) /t SE  
where, SEb and SEβ are the standard errors of the slope for the
TPL and Iwao’s model, respectively.
Calculated values are compared with tabulated (t) values
with n-1 degrees of freedom (Feng and Nowierski, 1992). If
the calculated t (tc)<t-table (tt), the null hypothesis (b = 1)
would be accepted and the spatial distribution would be
random. If tc>tt, the null hypothesis would be rejected and if b
> 1 and b < 1, the spatial distribution would be aggregated and
uniform, respectively (Naeimamini et al., 2014).
The annual data was pooled between the two years and
the overall distribution coefficients were used.
The presence or absence of differences between years was
calculated based on the following formulas:
where, b1 and b2 were Taylor’s coefficient for the two years,
β1 and β2 were Iwao’s coefficient for the two years, and SE1
and SE2 were their standard errors with (n1 + n2)-2 degrees of
freedom (Feng and Nowierski, 1992). The data of the two
years were integrated and a total distribution coefficient was
estimated only when the difference between the coefficients of
the two years was not significant.
Development of sampling plan (minimum sample size
number).
The coefficient from the IPR model was used to
determine the minimum sample size requirements for
estimating population means for each year with fixed levels of
precision. Precision (D) was defined as follows:
X
S
D
X



Where,
X
S  is the standard error of the mean and D is a fixed
proportion of the absolute mean of the population involved.
This is also known as the allowable error, or fixed precision
level, with which the mean is measured (Lindblade et al.,
2000). Estimators with standard errors of 5%, 10%, and 15%
at 0.05 probabilities were chosen for the present study.
The number of samples required to estimate the mean
with fixed precision was determined by solving the following:
2
n ( 1) / ( 1) /C       
Where, a and b are coefficients obtained from IPR. Because
Iwao’s regression coefficients provided the best explanation
for the data, sampling recommendations were developed based
on Kuno’s (1969) formula.
All obtained data were depicted graphically by Microsoft
Excel 2010.
III. RESULTS AND DISCUSSION
1- Population density of A. tubercularis on certain mango
cultivars:
Data represented in Table (1), revealed that the statistical
analysis of data indicated that, there was a highly significant
difference among mango cultivars to infestation by A.
tubercularis. It is clear from the results that the mean
maximum population of the pest was observed on Goleck
cultivar with an average of 92.51 ± 5.47 and 85.51 ± 4.28
individuals per leaf during the two years, respectively as
compared with the other tested cultivars. But, the cultivars of
Ewaise, Zebda, Hindi Bisinnara and Sediek were moderately
population of the pest with an average of 78.23 ± 4.63, 69.29 ±
3.57; 75.58 ± 4.01, 65.70 ± 2.92; 64.07 ± 3.44, 62.39 ± 3.01 and
59.08 ± 2.89, 51.00 ± 2.17 individuals per leaf during the two
years, respectively. However, the minimum individuals of the
population were recorded on Balady cultivar with an average
of 45.01 ± 2.60 and 37.72 ± 1.82 individuals per leaf through
the two years, respectively.
Statistically, highly significant differences among the six
tested cultivars regarding the level of infestation were
obtained (L.S.D. values were 6.12, 4.32 and 4.89) through the
first, second year, and between the two years, respectively. As
well, highly significant differences between the first and the
second years for A. tubercularis population (L.S.D. was
3.39).It is clear from the results that the mean numbers were
higher in the first year (69.08 ± 1.84 per leaf) than that of the
second year (61.98 ± 1.53 per leaf).
In general, it could be concluded that Goleck mango
cultivar was the most preference to white mango scale insect,
A. tubercularis, followed by Ewaise, then by Zebda and Hindi
Bisinnara, while the Balady cultivar was less preferable

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cultivar for this insect.
It is clear that the differences in the population densities
of A. tubercularis on different mango cultivars which may be
due to the differences not only in the environmental conditions
(such as temperature, relative humidity) but also there are
numerous other factors such as the leaf structure (density of
stomata, softness of tissues and size of leaves) as well as
growth features (such as growing period) for the tested
cultivars of mango.
We concluded that the host plant affects the development
of pest and that the choice of the most appropriate cultivar can
help to reduce pest infestation, and is therefore an additional
component to be included in the integrated pest management
of mango.
The degree of infestation on different mango cultivars
varied according to insect species. In this respect, Selim
(2002) in Egypt, however with different insect species, also
studied the susceptibility of five mango cultivars to infestation
with two armoured scale insect pest, Insulaspis pallidula and
Aonidella aurantii and recorded that I. pallidula infested all
cultivars more than A. aurantii. On the other hand, Hindy
cultivar was the most susceptible cultivar to infestation with
both scale insects followed by Mabrouka, then by Kobania
and Taimour, while the least susceptible cultivar was Dabsha.
Bakry (2009) in Egypt, however with different insect species,
also studied the variability among four mango cultivars in the
levels of infestation with two armoured scale insects, I.
pallidula and A. aurantii. He recorded that the population
density of A. aurantii was greater than that of I. pallidula. On
the other hand, grafted Balady was the most infested cultivar
with the two scale insects followed by Hindy, then by Goleck,
while the least infested cultivar was seedy Balady cultivar for
two-scale insects.
2. Sampling program:
The obtained values in Table (2) showed that the relative
variation (R.V.%) for the primary sampling data of A.
tubercularis indicated that the population densities of pest
ranging from (4.88 to 5.92%) and (4.26 to 5.15%)in the all
different mango cultivars through the first and second years,
respectively. As well, the R.V. (%) for the primary sampling
data of A. tubercularisindicated that the mean population
densities were 5.49, 4.69, and 4.78% during the first and
second years, and for the two years combined, respectively
(Table, 3). The values of R.V.% were very appropriate for a
sampling program. However, with different insect species and
different hosts, Naeimamini et al. (2014) stated that the
relative variation for the primary sampling data of different
stages of Pulvinaria floccifera (Hemiptera: Coccidae) was less
than 25% and were acceptable. Bakry (2018) reported that the
relative variation for the primary sampling data of total
populations of Waxiella mimosae (Signoret) (Coccomorpha:
Coccidae) on sunt trees ranged from 8.52 to 19.79% in all
seasons, as well as over the entire year.
Bakry (2020) recorded that the R.V. (%) for the primary
sampling data of Parlatoria oleae on mango trees indicated
that the total population density was 2.41, 2.35 and 1.73%
during the first and second years, and for the two years
combined, respectively. Bakry and Arbab (2020a) reported
that the relative variation for the primary sampling data of
Icerya seychellarum on guava trees indicated that total
population density was 4.07 (2017- 2018), 5.62 (2018-2019)
and 3.55% (pooled). Bakry and Shakal (2020b) recorded that
the relative variation (R.V.%) for the primary sampling data of
S. graminum indicated that the population densities of pest
ranging from (8.90 to 11.56%) and (7.21 to 10.28%)in the all
different cultivars and lines of wheat through the two growing
seasons, respectively. As well, the R.V. (%) for the primary
sampling data of S. graminum indicated that the mean
population densities was 9.78 and 8.04% during the first and
second growing seasons, respectively.
3-Spatial distribution:
3.1- Distribution indices:
The results in Tables (2 and 3) showed that the spatial
distribution among the sample units was determined by twenty
two indices of distribution. The distribution results using the
variance of A. tubercularis population on mango trees was
greater than the general average of the population densities by
the pest, and thus the variance-to-mean ratio (S2
/m) was
greater than one were recorded in the all tested mango
cultivars. Therefore, the spatial distribution of A. tubercularis
population was aggregated over the entire year and for the two
cumulative years.
The Lewis index of the pest was significantly greater than
the index of contagious dispersion. Similar conclusions were
made from the results of the Cassie index. The mean
population of the pest distribution was greater than zero;
therefore, A. tubercularison all tested mango cultivars had an
aggregated distribution. The K values of the negative binomial
distribution of A. tubercularispopulation ranged from 2 to 8 in
all mango cultivars during the first year, thus indicating
moderate aggregation, and the higher than 8 (K> 8) indicates a

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random population during the second year in the all tested
cultivars and on the cumulative analysis.
The Index values of mean clumping (IDM) of the pest in
all mango cultivars were positive for the negative binomial.
The Z-test values were greater than 1.96. The index of
patchiness was greater than one and Green’s index was greater
than zero. All these indices showed an aggregated distribution
for the population of A. tubercularis in all the different mango
cultivars during the two years and when the data were pooled.
The temporal changes in the spatial distribution pattern of
A. tubercularispopulation during each year were evaluated
using 1/k (the aggregation index). The value was greater than
one, thus indicating an aggregated pattern that became more
dispersed with time.
Also, the highest values for the variance to- mean, the
index of mean clumping, Z-test, patchiness index and green’s
index on the population density of A. tubercularis were
(15.51, 14.51, 28.54, 1.16 and 0.31) for the first year
(2017/2018) and (10.23, 9.23, 21.36, 1.11 and 0.20) during the
second year (2018/2019), were recorded on Goleck mango
cultivar as compared with the other tested mango cultivars
during the two years, respectively. This cultivar was suffered
the highest population density for this pest as mentioned
previously (Table, 2).
In contrary, the lowest values for the distribution
i.e.(S2
/m, IDM, Z test, x*/m, and GI) on the population of A.
tubercularis were (6.76, 5.76, 15.57, 1.10 and 0.12) were
recorded on Sediek mango cultivar than those the tested
mango cultivars during the first year of study (2017/2018),
respectively.
While, during the second year (2018/2019), the lowest
values for these indices (S2
/m, ID M, Z test and GI) were
recorded on Balady mango cultivar as compared with the other
mango cultivars was (4.22, 3.22, 10.26 and 0.07), respectively.
This cultivar was exhibited the lowest population density of A.
tubercularis as mentioned previously. However, the lowest
value of the index of patchiness (x*/m) by 1.07 was recorded
on Sediek mango cultivar (Table, 2).
The values of population aggregations (λ) were all less
than 2 in all tested mango cultivars over the entire year and for
the two cumulative years, however, indicating that the
aggregation phenomenon may be caused by environment
variations (Table, 2). A similar conclusion was found to occur
in distribution of Parapoynx crisonalis (Lepidoptera:
Crambidae) on water chestnuts plant (Li et al., 2017).
The results in Table (3) show that the values of
distribution indices of mean population density of A.
tubercularis were higher in the first year (2017/2018) as
compared to the second year (2018/2019). This evidence may
be due to the general average of population density A.
tubercularis was (69.08 ± 1.84 per leaf) during the first year
was higher than in the second one (61.98 ± 1.53 per leaf).
It is clear that the mango cultivars affect the population
density and spatial distribution of A. tubercularis. Therefore,
the spatial distribution for the population of A. tubercularis
using twenty two distribution indices indicated an aggregated
distribution in all different mango cultivars in the two
successive years and for the two cumulative years, that due to
the effects of environmental factors (Table, 2 and 3).
However, there is no study in the literature regarding the
distribution patterns of A. tubercularis. Studying different
insect species and different hosts, Chellappan et al. (2013)
reported that the value of mean crowding increased with an
increase in the mean population density of Paracoccus
marginatus (Hemiptera: Pseudococcidae). Li et al. (2017)
recorded that the K value of the negative binomial
distribution, aggregation index, and Cassie index were all
higher than zero during May. This would indicate that
Parapoynx crisonalis (Lepidoptera: Crambidae) larvae were in
an aggregated distribution. Bala and Kumar (2018) recorded
that the values of the Lewis index for all sampling dates of the
bug, Chauliops fallax (Hemiptera: Malcidae) population on
soybean were also found to be more than one, thus indicating
that the distribution of the bug population was aggregated.
Bakry (2018) studied the spatial distribution of W.
mimosae on sunt trees using 14 dispersion indices and
recorded that all the models exhibited an aggregated
distribution and followed a negative binomial distribution
pattern all the different life stages and for the total population
of W. mimosae over all seasons and for the pooled data over
the two years of study (2016 to 2018). Bakry (2020) recorded
that the spatial distribution of P. oleae on mango trees using
twenty one distribution indices and stated that all indices of
distribution indicated significant aggregated behaviour in each
year, except, the K values of the negative binomial distribution
of the total population ranged about 15-17 for each year
during the two successive years, indicating random behavior.
Bakry and Arbab (2020a) studied that the spatial distribution
of I. seychellarum on guava trees using distribution indices,
indicated aggregated behaviour over the entire year. Bakry
and Shakal (2020b) recorded that the spatial distribution
pattern of Schizaphis graminum (Hemiptera: Aphididae) on

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some wheat cultivars and lines. They found that all
distribution indices indicated a significant aggregated
behaviour during each growing season in all the tested wheat
cultivars and lines.
3.2- Regression methods:
Taylor’s power law regression showed highly significant
positive relationships between the log (mean of the
population) and log (variance) for the population densities of
A. tubercularis in the all different mango cultivars during the
first and second years of study (Table, 4 and Fig. 1).
The calculated regression coefficient from Taylor’s
method indicated that an increase of one degree in the log
(mean of the population) would increase the log of variance by
approximately 1.84 to 2.43 for the population of A.
tubercularis in the all tested mango cultivars during the first
and second years, and on the pooled data (Table, 4).
The regression coefficient values from TPL were
significant and greater than one. The values were (2.02, 1.87
and 1.95) during the first and second years and during the
cumulative years, respectively, and the values of t-calculated
of the slope (tc) >t-table (tt) in all mango cultivars during the
two years, respectively, indicated an aggregated distribution
for the population densities of A. tubercularis.
The relationship between these factors gave a good fit for
Taylor’s model and the R2
was 94.44, 95.63 and 93.14 for the
population of A. tubercularis in all mango cultivars during the
first and second years and on the cumulative years. The R2
value showed that the increase in values of population
variance occurred due to the increase in mean population
density.
In this model, the heterogeneity of the regression model
indicated that the slope of TPL did not differ significantly
between the two years (TPL: slope = 2.02, SEb = 0.15, and n1
= 12 for the first year and the slope = 1.87, SEb = 0.13, and n2
= 12 for the second year). The values of t-calculated (tc) = 0.77
<t-table (tt) = 2.07 when the df = [(n1+n2)-2] = 22 for the total
pest population between the two years.
Taylor (1984) showed that the slope (b) is an index of the
spatial distribution characteristic of the species; however,
other studies have shown that b is not species specific and
varies among environments and developmental stages
(Downing, 1986).
The regression method of Iwao described the relationship
between population mean ( X ) and mean crowding index
*
X
for the population of A. tubercularisin all tested mango
cultivars for the first and second years and during the two
cumulative years. The regression coefficient (β) values were
significantly greater than one, was 1.02, the values of t-
calculated of slope (tc) > t-table (tt) for the population of pest
in the all tested mango cultivars and all years and on the two
cumulative years, indicated an aggregated distribution for the
population of A. tubercularis (Table, 4).
The calculated regression coefficient of Iwao’s method
indicated that an increase of one degree in the population
mean would increase the mean crowding index approximately
one degree for the population in all tested years.
Additionally, the intercept values (α) or the index of basic
contagion were negative and < zero and > -1. The values of α
ranged from -0.83 to -2.32 for the pest population in both
years as well as for the cumulative years (Table, 4). The
negative values indicated that aggregation was from
individuals rather than from colonies and the smaller than zero
values indicated that of the pest population, the basic
component of the population tended to be a single individual
in both years and for the two cumulative years.
The relationship between the population mean and mean
crowding index had a better fit. R2
was 99.9% for the pest
population in all tested years as well as the pooled data. R2
showed that the increase in the mean crowding index occurred
due to the increase in the population mean (Table, 4 and Fig.,
1).
The heterogeneity of the regression model showed that
the slope of the IPR did not differ significantly between the
two years: IPR: slope = 1.02, SEb = 0.004, and n1 = 12 for the
first year and slope = 1.02, SEb = 0.002, and n2 = 12 for the
second year. The t-calculated (tc) values = 1.36 <t-table (tt) =
2.07 when the df = [(n1+n2)-2] = 22 for the total pest
population between the two studied years. These findings are
in agreement with those reported by Tonhasca et al. (1996)
who recorded that patchiness regression is not subject to the
stabilizing effect of a log transformation.
The present results revealed that the regression coefficient
(b) values of Taylor and (β) values of Iwao were both
significantly greater than one and the t-calculated slope values
(tc) >t-table (tt). Therefore, the population of A. tubercularis
tended to have an aggregated distribution.
Generally, the regression models of TPL and Iwao are
used to estimate the spatial distribution of a pest. The results
from the present study showed an aggregated pattern and a
negative binomial distribution pattern for the population
density of A. tubercularis. Factors that may influence the

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spatial distribution of pests include the level of infestation,
environmental condition, or behavioural characteristics of the
insects. In addition, these two regression methods were more
accurate and produced different results than the other
distribution indices used in the present study. These models
were based on sample means and population variances.
The present results are generally in agreement with those
of Hsu et al. (2001) however with the same genus of insect
and different host, they studied that the dispersion pattern of
varied stages generated by Iwao’s patchiness regression and
Taylor’s power law for A. yabunikkei in different sampling
units of camphor trees, Cinnamomum camphora (L.), were
compared. Taylor's power law provided a consistently good fit
to the data, whereas the fit of Iwao’s patchiness regression was
erratic, and the values of aggregation index of Taylor’s power
law (1.76 to 2.65) were narrower than those of Iwao’s (1.26 to
11.83), but both indices (b > 1.7 and β > 1.2) indicate a
clumped distribution pattern in all sampling units. Also, Lee et
al. (2005) however with the different insect species and
different hosts, reported an aggregated distribution pattern for
leaf miner in tomato greenhouses, because all b values from
Taylor's power law regressions were significantly greater than
one.Furthermore, aggregated distribution patterns of other
pests using these aggregation indices have been reported
previously (Naseri et al., 2009; Payandeh et al., 2010; Beltra
et al., 2013; Nematollahi et al., 2014; Arbab, 2014; Arbab
and McNeill, 2014 and Bakry and Arbab 2020a).
4- Optimum number of sample size:
The optimal sample size number was calculated using
Iwao’s regression coefficient, and the relationship between the
optimum sample size number and mean numbers of A.
tubercularispopulation with levels of precision of 5%, 10, and
15% were calculated during the first and second years and for
the two cumulative years (Table, 5 and Fig. 2).
The minimum number of samples that were required were
shown to decrease rapidly with an increase in mean population
density of A. tubercularis (inverse relationship) at all precision
levels (5%, 10%, and 15%, Fig. 2).
These results agree with those obtained by Arbab and
McNeill (2014); however, they used different insect species
and host and reported that the optimum sample size number
decreased rapidly as Sitona humeralis weevil density
increased at the 10% and 15% precision levels. Bakry and
Arbab (2020a) recorded that minimum numbers of sample
required decreased rapidly with increased mean total
population density of I. seychellarum at levels of precision of
(5, 10 and 15%).
The optimum sample size fluctuated during each year of
study and depended upon the population density of A.
tubercularisand the desired precision level. At the 15%
precision level, the mean number of samples required for
estimating the population density of A. tuberculariswas
approximately three leaves in each of the years, which would
be suitable for integrated pest management purposes.
However, the values used for population ecology studies
require a precision of 10%, thus the sample size was increased
and was approximately 6 leaves during each year for the two
successive years (Table, 5 and Fig. 2). However, for a more
accurate estimate (i.e. 5%), the required sample size of mango
leaves increased to approximately 22 leaves in the first and
second years, respectively (Table, 5 and Fig. 2).
The results showed that the sample size required to
achieve a desired level of precision was similar from one year
to another. This could be explained by the fact that the
population pest density from one year to another was similar
and may be due to the environmental conditions, which were
suitable during both years of study.
The different precision levels (5%, 10%, and 15%)
adopted in the present study could be chosen for ecological or
insect behavioural studies. This study showed that precision
levels of 5% and 10% are suitable for ecological or insect
behavioural studies where a higher level of precision is
required, whereas for pest management programmes, a 15%
level is acceptable. However, even though such a number of
samples is acceptable for research purposes, it is not
practicable for agronomic or pest management.
These results agree with those obtained by Arbab and
McNeill (2014). One attribute of the optimum sample size is
that, at a given density, the required sample size increases as
the target population becomes more aggregated (Wipfli et al.,
1992). Successful management of A. tubercularisstrongly
depends on the development of an appropriate sampling plan
(i.e. easy to implement, suitable for rapid decision-making
processes). In sampling programmes, precision and cost-
effectiveness are two of the most important factors that must
be considered (Pedigo, 1994). The development of a
sequential sampling scheme with a fixed statistical precision,
therefore, may be useful for estimating the density of A.
tubercularisin mango orchards. Such an estimation, in turn,
would be valuable for ecological and pest management
studies. A sampling programme can be used in ecological

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investigations (Faleiro et al. 2002) and when detecting pest
levels that lead to a justification of control measures (Arnaldo
and Torres 2005).
Table.1: Mean numbers of A. tubercularis population on different mango cultivars during the two successive years of
(2017/2018 and 2018/2019):
Means followed by the same letter (s), in each column, are not significantly different at 0.05 level probability, by Duncan's
multiple range test (DRMT).
Table.2: Estimated parameters for spatial distribution of population density of A. tubercularis on different mango cultivars
during the two successive years (2017/2018 and 2018/2019).

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Therefore, the distribution pattern of the insect is
essential for the management of A. tubercularis. The next
steps are to develop an efficient scouting programme and
establish threshold densities for action that will inform
growers on when the pest is active and when interventions
(e.g. insecticides) can be applied. The results from the
present study are indicative of populations present in the
field and provide information on the relative change over
time. Thus, the spatial distribution parameters of this species
can be employed to estimate the population density of A.
tubercularis. For the monitoring, sampling, and population
density estimation of A. tubercularis, the spatial distribution
pattern should be considered because the minimal sample
size is dependent on the spatial pattern of the sampled
population.
Table.3: Estimated parameters for spatial distribution of mean population density of A. tubercularis during the two successive
years (2017-2019).

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Table.4: Parameters estimation for spatial distribution of A. tubercularis population on different mango cultivars derived from
different regression methods during the two successive years (2017/2018 and 2018/2019).
Table.5: Maximum, minimum and mean population density for the population density of A. tubercularis and optimum sample size
(n) for achieving a fixed precision levels of (D = 0.05, 0.10 and 0.15) using enumerative sampling procedures during
the two successive years (2017/2018 and 2018/2019).

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Fig. 1: Regression analyses of Taylor's power law (A) and Iwao's patchiness (B) for A. tubercularis population on mango trees
during the two successive years (2017/2018 and 2018/2019).

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Fig. 2: Relationship between population density for the population density of A. tubercularis and optimum number of sample size
(n) for achieving a fixed precision levels of (D = 0.05, 0.10 and 0.15) using enumerative sampling procedures during the two
successive years (2017/2018 and 2018/2019).
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Examining the spatial distribution pattern and optimum sample size for monitoring of the white mango scale insect, Aulacaspis tubercularis (Newstead) (Hemiptera: Diaspididae) on certain mango cultivars